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Rayleigh–Bénard instability generated by a diffusion flame

Published online by Cambridge University Press:  08 November 2013

P. Pearce  and
J. Daou
P. Pearce
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK
J. Daou*
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK
*
†Email address for correspondence: joel.daou@manchester.ac.uk
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Abstract

We investigate the Rayleigh–Bénard convection problem within the context of a diffusion flame formed in a horizontal channel where the fuel and oxidizer concentrations are prescribed at the porous walls. This problem seems to have received no attention in the literature. When formulated in the low-Mach-number approximation the model depends on two main non-dimensional parameters, the Rayleigh number and the Damköhler number, which govern gravitational strength and reaction speed respectively. In the steady state the system admits a planar diffusion flame solution; the aim is to find the critical Rayleigh number at which this solution becomes unstable to infinitesimal perturbations. In the Boussinesq approximation, a linear stability analysis reduces the system to a matrix equation with a solution comparable to that of the well-studied non-reactive case of Rayleigh–Bénard convection with a hot lower boundary. The planar Burke–Schumann diffusion flame, which has been previously considered unconditionally stable in studies disregarding gravity, is shown to become unstable when the Rayleigh number exceeds a critical value. A numerical treatment is performed to test the effects of compressibility and finite chemistry on the stability of the system. For weak values of the thermal expansion coefficient $\alpha $, the numerical results show strong agreement with those of the linear stability analysis. It is found that as $\alpha $ increases to a more realistic value the system becomes considerably more stable, and also exhibits hysteresis at the onset of instability. Finally, a reduction in the Damköhler number is found to decrease the stability of the system.

JFM classification

Convection: Buoyancy-driven instability Reacting Flows: Combustion Reacting Flows: Flames
Type
Papers
Information
Journal of Fluid Mechanics , Volume 736 , 10 December 2013 , pp. 464 - 494
Copyright
©2013 Cambridge University Press 

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References

Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large-scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81 (2), 503–537.Google Scholar
Arai, M., Sato, H. & Amagai, K. 1999 Gravity effects on stability and flickering motion of diffusion flames. Combust. Flame 118 (1), 293–300.Google Scholar
Bénard, H. 1900 Les tourbillons cellulaires dans une nappe liquide. Rev. Gen. Sci. Pures Appl. 11 (1309), 1261–1271.Google Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32 (1), 709–778.Google Scholar
Bormann, A. S. 2001 The onset of convection in the Rayleigh–Bénard problem for compressible fluids. Contin. Mech. Thermodyn. 13 (1), 9–23.CrossRefGoogle Scholar
Boussinesq, J. 1903 Theorie Analytique de la Chaleur. Gauthier-Villars.Google Scholar
Buckmaster, J. & Jackson, T. L. 2000 Holes in flames, flame isolas, and flame edges. Proc. Combust. Inst. 28 (2), 1957–1964.Google Scholar
Buckmaster, J. D. & Ludford, G. S. S. 1983 Lectures on Mathematical Combustion. SIAM.Google Scholar
Buckmaster, J., Nachman, A. & Taliaferro, S. 1983 The fast-time instability of diffusion flames. Physica D 9 (3), 408–424.CrossRefGoogle Scholar
Buckmaster, J. & Peters, N. 1988 The infinite candle and its stability – A paradigm for flickering diffusion flames. Symp. (Intl) Combust. 21 (1), 1829–1836.Google Scholar
Bychkov, V., Modestov, M., Akkerman, V. & Eriksson, L. E. 2007 The Rayleigh–Taylor instability in inertial fusion, astrophysical plasma and flames. Plasma Phys. Control. Fusion 49 (12B), B513.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Instability. Dover.Google Scholar
Cheatham, S. & Matalon, M. 1996a Heat loss and Lewis number effects on the onset of oscillations in diffusion flames. In Symp. (Intl) Combust., 26, pp. 1063–1070.Google Scholar
Cheatham, S. & Matalon, M. 1996b Near-limit oscillations of spherical diffusion flames. AIAA J. 34 (7), 1403–1409.Google Scholar
Cheatham, S. & Matalon, M. 2000 A general asymptotic theory of diffusion flames with application to cellular instability. J. Fluid Mech. 414, 105–144.Google Scholar
Daou, J. & Al-Malki, F. 2010 Triple-flame propagation in a parallel flow: an analytical study. Combust. Theor. Model. 14 (2), 177–202.Google Scholar
Echekki, T., Chen, J. Y. & Hegde, U. 2004 Numerical investigation of buoyancy effects on triple flame stability. Combust. Sci. Technol. 176 (3), 381–407.Google Scholar
Fröhlich, J., Laure, P. & Peyret, R. 1992 Large departures from Boussinesq approximation in the Rayleigh–Bénard problem. Phys. Fluids A 4, 1355.CrossRefGoogle Scholar
Gelfgat, A. Y. 1999 Different modes of Rayleigh–Bénard instability in two- and three-dimensional rectangular enclosures. J. Comput. Phys. 156 (2), 300–324.Google Scholar
Getling, A. V. 1998 Rayleigh–Bénard Convection: Structures and Dynamics. World Scientific.Google Scholar
Jiang, X. & Luo, K. H. 2000 Combustion-induced buoyancy effects of an axisymmetric reactive plume. Proc. Combust. Inst. 28 (2), 1989–1995.CrossRefGoogle Scholar
Kim, J. S. 1997 Linear analysis of diffusional-thermal instability in diffusion flames with Lewis numbers close to unity. Combust. Theor. Model. 1 (1), 13–40.Google Scholar
Kim, J. S., Williams, F. A. & Ronney, P. D. 1996 Diffusional-thermal instability of diffusion flames. J. Fluid Mech. 327, 273–301.Google Scholar
Kirkbey, L. L. & Schmitz, R. A. 1966 An analytical study of the stability of a laminar diffusion flame. Combust. Flame 10 (3), 205–220.Google Scholar
Kukuck, S. & Matalon, M. 2001 The onset of oscillations in diffusion flames. Combust. Theor. Model. 5 (2), 217–240.Google Scholar
Liñán, A. 1974 The asymptotic structure of counterflow diffusion flames for large activation energies. Acta Astronaut. 1 (7), 1007–1039.Google Scholar
Lohse, D. & Xia, K. Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335–364.Google Scholar
Majda, A. & Sethian, J. 1985 The derivation and numerical solution of the equations for zero Mach number combustion. Combust. Sci. Technol. 42 (3–4), 185–205.Google Scholar
Matalon, M. 2009 Flame dynamics. Proc. Combust. Inst. 32 (1), 57–82.Google Scholar
Matalon, M. & Metzener, P. 2010 The effect of thermal expansion on diffusion flame instabilities. J. Fluid Mech. 647, 453–472.Google Scholar
Matkowsky, B. J. & Sivashinsky, G. I. 1979 An asymptotic derivation of two models in flame theory associated with the constant density approximation. SIAM J. Appl. Maths 37 (3), 686–699.Google Scholar
Metzener, P. & Matalon, M. 2006 Diffusive-thermal instabilities of diffusion flames: onset of cells and oscillations. Combust. Theor. Model. 10 (4), 701–725.Google Scholar
Miklavčič, M., Moore, A. B. & Wichman, I. S. 2005 Oscillations and island evolution in radiating diffusion flames. Combust. Theor. Model. 9 (3), 403–416.Google Scholar
Paillere, H., Viozat, C., Kumbaro, A. & Toumi, I. 2000 Comparison of low Mach number models for natural convection problems. Heat Mass Transfer 36 (6), 567–573.Google Scholar
Rayleigh, Lord 1916 On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Phil. Mag. 32 (192), 529–546.Google Scholar
Rehm, R. G. & Baum, H. R. 1978 The equations of motion for thermally driven, buoyant flows. J. Res. Natl Bur. Stand. 83 (3), 297–308.Google Scholar
Shampine, L. F., Kierzenka, J. & Reichelt, M. W. 2000 Solving boundary value problems for ordinary differential equations in MATLAB with bvp4c. Tech Rep. The MathWorks.Google Scholar
Sivashinsky, G. I. 1977 Nonlinear analysis of hydrodynamic instability in laminar flames – i. Derivation of basic equations. Acta Astronaut. 4 (11), 1177–1206.Google Scholar
Sohn, C. H., Chung, S. H. & Kim, J. S. 1999 Instability-induced extinction of diffusion flames established in the stagnant mixing layer. Combust. Flame 117 (1), 404–412.Google Scholar
Vance, R., Miklavcic, M. & Wichman, I. S. 2001 On the stability of one-dimensional diffusion flames established between plane, parallel, porous walls. Combust. Theor. Model. 5 (2), 147–161.CrossRefGoogle Scholar
Wu, K. K. & Chen, C. H. 2003 A numerical analysis of ignition to steady downward flame spread over a thin solid fuel. Combust. Sci. Technol. 175 (5), 933–964.Google Scholar